# The space oblique conic projection.

IntroductionSide-looking radars (SLRs) have enjoyed continued success in mapping the earth's phenomena--including geology, vegetation, forestry, and topography--since the 1960s from aircraft, and from the 1970s from space platforms (Simonett 1970; Leberl 1976; Henderson and Lewis 1998). Recent attention has turned to mapping terrain, which can be achieved with radar in all weather and illumination conditions, and to high resolution methods such as interferometric SAR. Nevertheless the string of successful radar satellites (SEASAT, SLAR, RADARSAT, TerraSAR-X, Magellan, SIR-C, etc.) means that radar data for earth are ubiquitous, and now are available over a 50-yar timespan. Work, to date, on mapping from SAR has been based on single images, stereo, and image overlap (Leberl 1976). Most solutions for an earth geometry for small-scale mapping have used ground truth and rubber-sheeting style geometric rectifications. In this paper, we seek a purely analytic solution based on orbital and radar geometry, and on map projections, in particular the class of map projections known as space projections (Ren and Zhu 2001; Liucheng 2003).

Space map projections are dynamic map projections specifically established for a remotely sensed satellite platform, in which both geometry and time play a part in determining the structure of the image. It is desirable to specify the geometry of a map projection model so that satellite imagery can be geo-rectified to earth-based coordinate systems for cartography and further image applications. A space map projection is necessary to establish the precise processing and cartography of SLR imagery, in particular from Synthetic Aperture Radars that generally image at a fixed angle to the vertical. To date, few have considered the geometry of SLR, and fewer still have considered a direct map projection of the data. A simple view of the radar imaging geometry is shown in Figure 1.

The concept of space projections is generalized from the initial work of Colvocoresess, who pioneered the field while with cartographic coordinator for earth satellite mapping at the United States Geological Survey in 1974 with the space oblique Mercator projection (SOM) (Colvocoresses 1974). Since then, other space projections have been developed and the SOM equations derived (Junkins and Turner 1977; Snyder 1981). More recently, work on space projections has been focused in China (Yang, Snyder, and Tobler 1999; Liucheng 2003; Liucheng et al. 2010).

Side-looking radar (SLR) is an active microwave remote-sensing system: the imagery data are obtained by sending a radar pulse toward the ground, then receiving the reflected pulse wave after its interactions with the ground surface. The mathematical foundation for the system geometry is an important problem for cartographers to solve, so that the imagery can be precisely geometrically corrected, and the errors minimized. The geometric distortion of SLR imagery can be influenced by many factors, such as the orientation, roll, pitch and yaw of the satellite, the rotation and curvature of the earth, the orbital precession of the satellite and the selection of the image projection plane, among many other factors. The selection of a suitable map projection is very important for high precision and accurate rectification of SLR imagery.

Existing mathematical models for an SLR representation are based on instantaneous imaging equations which, in turn, are built point by point or row by row for the scanner's sweep across the earth's surface (Zhengbo, Jingyu, and Guochao 1991). However, these models cannot represent all of the image data, which is obtained continuously by the radar. The Space Oblique Mercator projection, the Conformal Space Projection (CSP) (Cheng 1996) and other space map projections have all been designed specifically for satellite imagery that is scanned around a nadir ground track, such as SPOT and Landsat imagery. The precision of the projection can only satisfy the demands of cartography within [+ or -] 1[degrees] of latitude and longitude around the ground track (Snyder 1981). However, the central line of the swath for SLR is about 270 km away from the ground track of the satellite generally, outside the region of [+ or -] 1[degrees] around the ground track of the satellite. For example, the effective imaging region of the SLR imaging system of SEASAT-A was [+ or -] 2[degrees] away from the ground track of the satellite (Leberl 1981). Obviously, the SOM and CSP projections are not suitable for representing SLR imagery, because the precisions of the projections do not satisfy cartographic accuracy requirements (Dowman and Peacegood 1989). Our solution to this problem is to design a space map projection specifically for the representation of the SLR image data.

The aim of this paper is to describe the space oblique conic (SOC) projection for SLR image data based on simulating the physical processes and geometric relations of imaging with SLR. A space projection has the characteristic of establishing the corresponding relation between pixels and ground points approximately while keeping the central line of the SLR swath distortion-free. SOC is a new time-related projection designed for the precise rectification of a satellite-based SLR.

Geometric principles of the space oblique conic projection

While accounting for the factors of satellite geometry and movement, for the rotation of the earth and for the satellite's orbital procession, assume a cone that is defined by a circular orbit (Figure 1) with the projection surface tangent to the spheroid of the earth, and for which the tangent line is the central line of a SLR swath. There exists a set of relative movement relations among the cone, the satellite and the earth, making up four principal motions: 1. the satellite's scanner sweep across the earth's surface; 2. the satellite's orbit; 3. the earth's rotation; and 4. the earth's orbital precession. To keep these motions from distorting the SLR image, the conic surface of the projection is made to oscillate along its axis at a compensatory rate that varies with latitude. Simulating the physical process of SLR imaging according to this model, a new projection is derived as follows. In order to solve the projection model, two conditions must be assumed: 1. zero length distortion on the central line of the image swath; and 2. Conformity, or local "shape" preservation.

Because the SOC projection is a periodic function of time t, its formula can be denoted as

x = [f.sub.1] ([phi], [lambda], t), y = [f.sub.2] ([phi], [lambda], t)

Ground track of the central line for a side-look region

In order to establish the formula for the SOC, first the projection formula of the ground track projection of the central line of the side-look region must be established.

Denoting the central line as L. Assuming A([[phi].sub.L], [[lambda].sub.L], t) is any point on L, the corresponding projection on the projection plane is B([x.sub.L], [y.sub.L], t).

Maintaining length

Suppose the arc length of the central line L at time t is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

The corresponding arc length on the projection plane is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [v.sub.L](t) is the instantaneous velocity of the scanning point at time t on the central line L of the side-look region. Note that 0 < [tau] < t, where r is the time of the satellite perigee. In order to preserve length, that is s = s', then

[v.sup.2.sub.L] = [(d[x.sub.L]/dt).sup.2] + [(d[y.sub.L]/dt).sup.2] (3)

Maintaining curvature

In order to preserve the shape of the projected region, the curvature radius of the central line L must be equal to the instantaneous curvature radius of its projection [rho](t) in the projection plane, that is:

[([d.sup.2][x.sub.L]/d[s.sup.2]).sup.2] + [([d.sup.2][y.sub.L]/d[s.sup.2]).sup.2] = 1/[[rho].sup.2](t) (4)

Combining Formulas (3) and (4), the equations of the ground track projection can be solved as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Therefore, the projection of the central line L of each side-look region is (Figure 2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The space oblique conic projection formula

In this section, we present the derivation of the map projection approximately in the manner it was developed and the formula will be deduced based on the projection formula of the central line of the side-looking region already presented. Since the satellite orbit is generally oblique to the equator, a transformation at the equator is needed in order to use the static conformal conic projection formula expediently (Yang, Snyder, and Tobler 1999).

With reference to Figure 3, we consider the projection of all sensed points within a finite region on the reference ellipsoid, centered on the ground track onto the map plane. This problem is approached using the simple notion that very small displacements are made on the ellipsoid from a locally very-nearly straight line. Accordingly, displacements near the ground track from nearby points might be well-approximated by displacements near the equator of an oblique conic projection (where the oblique equator is locally tangent to the ground track). The resulting map projection (based on this approximation) is developed such that conformity and length preservation are rigorously satisfied only along the ground track, but the approximation remains accurate within several hundred kilometers of the satellite ground track.

Assuming zero Doppler effect, when the satellite orbits at instant t, the geometric relation of the conformal conic projection that is tangent to the central line of the side-looking region is shown as in Figure 4, and the angle between the direction of satellite orbit and the x axis is:

[epsilon] = arctan([[??].sub.L]/[[??].sub.L]) (7)

In Figure 3, rotating the instantaneous Cartesian geometry [x.sub.1]A[y.sub.1] to an angle [epsilon], we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

According to Wu and Yang (1989), taking any point A on the central line as the origin point and the direction of a satellite track as the [x.sub.1] axis, we can create a dynamic Cartesian system, as shown in Figure 4 - S is the satellite position here. Suppose B([phi]', [lambda]') is an arbitrary point in the side-looking region, then the conformal conic projection corresponding to the new equator is:

d = f([phi]') = C/[U.sup.[alpha]], [delta] = [alpha][lambda]' (9)

[x.sub.1] = d sin [delta], [y.sub.1] = [d.sub.0] - d cos [delta] (10)

where: [d.sub.0] = R cot [[phi]'.sub.0], C = R cot [[phi]'.sub.0][U.sup.[alpha].sub.0], [alpha] = sin [[phi]'.sub.0], U = tan ([pi]/4 + [lambda]'/2), [[phi]'.sub.0] = [s.sub.0]/R ([s.sub.0] is the distance from A to the ground track at time t), [delta] = [alpha][lambda]', R is the earth's radius, and [[phi]'.sub.0] is defined as by Snyder (1981) as the time of the ascending node.

Folding the coordinate system [x.sub.2]A[y.sub.2] with xoy together, as in Figure 3, for any point B in the sidelook region, its projection coordinates in the xoy system are:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where: t = the orbital time since the start; [v.sub.L](t)= the instantaneous velocity of the point A on the central line of the side-look region corresponding to the earth at the time instant t; and [rho](t) = the instantaneous curvature radius at point A on the central line; thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Latitude [phi](t) and longitude [lambda](t) of side-looking point B at time t

Assuming a side-look point is B([phi], [lambda]) along the satellite orbit at the instant t--that is the beam points to B([phi], [lambda])--then the geographic coordinate of the corresponding point [B.sub.0] on the central line of the side-look region is ([[phi].sub.0](t), [[lambda].sub.0](t)). With reference to Figure 5, assuming zero Doppler effect, B([phi], [lambda]) and [B.sub.0]([[phi].sub.0], [[lambda].sub.0]) are located in the plane, which is determined by the unit normal vector [bar.n] and the pulse vector [??], that is, the identity equation is satisfied:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

(z cos [phi] sin [lambda] - y sin [phi]) (cos [phi] cos [lambda] - cos [[phi].sub.0] cos [[lambda].sub.0]) + (x sin [phi] - z cos [phi] cos [lambda])(cos [phi] sin [lambda] - cos [[phi].sub.0] sin [[lambda].sub.0]) + (y cos [phi] cos [lambda] - x cos [phi] sin [lambda]) (sin [phi] - sin [[phi].sub.0]) = 0 (15)

Similarly, formula (15) is also true for the transformation of geographic coordinates [phi]', [lambda]' and [[phi]'.sub.0], [[lambda]'.sub.0]. According to Snyder (1981) and assuming zero Doppler effect, [lambda]' = [[lambda]'.sub.0] = 2([pi])t/[P.sub.2], [P.sub.2], [P.sub.2] is the period of the satellite orbit, and [[phi]'.sub.0] = [s.sub.0]/R are known. Substituting [lambda]', [[lambda]'.sub.0] and [[phi]'.sub.0] into Equation (15), the relation of [phi]' at time t can be solved. Then substituting [[phi]'.sub.0], [[lambda]'.sub.0] and [phi]', [lambda]' into the spherical triangle formula respectively, [[phi].sub.0], [[lambda].sub.0] and [phi], [lambda] can be computed using iterative methods.

Inverse transformation

Selecting the initial latitude and longitude [[phi].sup.(0)] = [[phi].sub.L](t), [[lambda].sup.(0)] = [[lambda].sub.L](t) for any timer, and substituting [[phi].sup.(0)] and [[lambda].sup.(0)] into Equations (11) and (12), then the corresponding coordinates [x.sup.(0)], [y.sup.(0)in] the SOC can be obtained. Constructing the iterative formula:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)

The matrix of partial derivatives can also be obtained from Equations (11) and (12). The coordinates [x.sup.(k)], [y.sup.(k)] of the SOC for the kth step can be obtained by substituting [[phi].sup.(k)], [[lambda].sup.(k)] into Equations (11) and (12) (Butcher 2003). Then [phi](t), [lambda](t) can be obtained with a relatively small numbers of iterations, according to Equation (16).

Projection distortion analysis

Taking the partial derivatives of Equations (11) and (12) yields:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

To compact the notation, we make use of the symbol "[phi] [right arrow] [lambda], t" whenever the identical equation for 2 or t results by simply replacing [phi] by [lambda] or t.

From Equation (15):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

SOC projection distortion

Ratio of meridian

[m.sup.2] = [E.sub.1] + [E.sub.3] [(dt/d[phi]).sup.2] + 2[F.sub.2](dt/d[phi])/[M.sup.2]

Ratio of latitude

[n.sup.2] = [E.sub.2] + [E.sub.3] [(dt/d[lambda]).sup.2] + 2[F.sub.3] (dt/d[lambda])/[r.sup.2]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Angular distortion

The tangent of the angle between latitude and longitude is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The distortion of azimuth angle after projection is:

cot [alpha]' = S/[H.sub.1]Mr tan [alpha] + [H.sub.3][M.sup.2] [tan.sup.2] [alpha](dt/d[lambda])

where

S = [E.sub.1][r.sup.2] + [F.sub.1]Mr tan [alpha] + 2[F.sub.2]Mr tan [alpha](dt/d[lambda]) + [F.sub.3][M.sup.2] [tan.sup.2] [alpha](dt/d[lambda]) + [E.sub.3][M.sup.2] [tan.sup.2] [alpha][(dt/d[lambda]).sup.2].

Example application

In order to get an SOC projection, we take the imaging system of SEASAT as an example. SEASAT was launched on 27 June 1978 (See: http://en.wikipedia.org/wiki/SEASAT) into a nearly circular 800 km orbit with an inclination of 108[degrees], and operated for 105 days until 10 October 1978. The coordinates of SOC and its inverse can be calculated using a simulated SAR orbital data. With reference to Leberl (1981), the relevant parameters of SEASAT are: satellite orbital period [P.sub.2] = 100 min; satellite orbit is a circle; obliquity angle of the orbital plane i = 108[degrees]; [omega] = 0.0; [OMEGA] = 0.0; orbital height H = 800 km; The bandwidth of side-looking strip D = 100km (Figure 6); view angles [[alpha].sub.1] = 16.9[degrees] to [[alpha].sub.2] = 23.1[degrees] (Figure 7); and distance from the central line to the ground track [s.sub.0] = 292km. Suppose the initial moment of satellite movement is at the descending node, selecting [t.sub.0] = [tau] = 0, [[theta].sub.0] = 0; Using an earth radius of R = 6378160.0m.

According to the data noted above, the pitch angle can be solved as [alpha] = [s.sub.0]/H = 20.9[degrees], the average angular velocity of the satellite [bar.n] = 2[pi]/[P.sub.2] = 3.6[degrees] = 0.062832 radian/min, the transformation latitude of the central line of the side-look region [[phi]'.sub.(0)] = [s.sub.0]/R = 2.623[degrees], the eccentric anomaly E = [bar.n]t at the instant t, the rotation angle of the earth [theta] = [[omega].sub.e]t.

Regarding of the rotation of the earth, the coordinates of the satellite can be obtained as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

According to Section 3, the corresponding formula for the SOC can be obtained:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting the result above into Equations (11) and (12):

x = 9.780451t + 3.211746 sin [pi]t/[P.sub.2] - 0.034421 sin 2[pi]t/[P.sub.2] - 0.006059 sin 4[pi]t/[P.sub.2] + 0.000648 sin 6[pi]t/[P.sub.2] + 0.000002 sin 8[pi]t/[P.sub.2] + d cos([delta] + [epsilon]) + [d.sub.0] sin [epsilon] y = [y.sub.0] -485.263782 cos [pi]t/[P.sub.2] -4.350619 cos 3[pi]t/[P.sub.2] -0.001143 cos 5[pi]t/[P.sub.2] + 0.00022 cos 7 [pi]t/[P.sub.2] + d sin ([epsilon] + [delta]) + [d.sub.0] cos [epsilon]

where the expressions of [epsilon] d, [d.sub.0], [delta] is Equation (13).

Conclusion

In order to satisfy the demands of cartography from an SLR imagery, the space map projection problem for SLR has been examined. The Space Oblique Conic projection has been specifically designed for this kind of imagery. The geometric model and the imaging process have been simulated, by conceiving a cone that is tangent to the central line of the side-look region instantaneously. According to the space geometric relations of the satellite orbit, earth rotation and the SOC, the sub-point projection of the central line of the side-look region has been developed, and the formula for the SOC has been derived. The research contributions here include the equatorial transformation, the SOC projection, the inverse transformation and a distortion analysis. As an example, image data for the SEASAT satellite has been used to establish the framework for SOC, and the coordinates in the SOC have been calculated.

A new method of space map projection has been proposed. Space map projections already established are the Space Oblique Mercator, the space azimuth projection based on the tangent plane as the projection plane, and the space cylindrical projection based on a cylindrical surface as the projection plane. In this paper, we establish a space projection based on a cone as the projection plane, the SOC. The forward and inverse formulas of SOC have been derived, adding a new space map projection model to the theory of space map projection.

With this contribution, a more suitable dynamical mathematical foundation has been established for the satellite-based SLR imagery. We hope that this direct projection is of value for cartography based on remote sensing and for the precise rectification of SLR imagery, both past and future.

http://dx.doi.org/10.1080/15230406.2013.810804

Acknowledgment

This research was funded by the National Natural Science Foundation of China (No. 41071287)

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Liucheng Ren (a), Keith C. Clarke (b) *, Chenghu Zhou (c) and He Ren (d)

(a) Institute of Remote Sensing Applications, Chinese Academy of Sciences, and Air Force Command College, Beijing, China; (b) Department of Geography, University of California, Santa Barbara 93106-4060, CA, USA, (c) State Key Laboratory of Resources & Environmental Information Systems, Institute of Geography, Chinese Academy of Sciences, Beijing 100101, China; (d) School of Science, School of Electronic and Information Engineering, Beijing Jiaotong University, Beijing 100044, China

* Corresponding author. Email: kclarke@geog.ucsb.edu

(Received 30 October 2012; accepted 4 February 2013)

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Author: | Ren, Liucheng; Clarke, Keith C.; Zhou, Chenghu; Ren, He |
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Publication: | Cartography and Geographic Information Science |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Sep 1, 2013 |

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